You are correct in your observation. However, the party switch happened over a 100 year period, not all at once. I will try to explain this shift, but it will be very oversimplified.
1860-1900
In the 1800s, the Republican Party was the party of the North. Because of this, they were against slavery. However, after the Civil War, voters stopped caring about civil rights as a political issue. The Republicans soon became a vehicle for Northern big business. The party switched its focus from civil rights to raising tariffs, creating a gold standard currency, and expanding overseas imperialism. During this period, the Republicans were the dominant party, winning most national elections. The Democrats formed a coalition of everyone who disliked the Republicans. Their base was in the Southern States, where the Civil War was seen as an invasion of the South by a Republican president. The Democrats also reached out to poor Catholic immigrants in the Northern cities, who disliked many aspects of the Protestant Republican agenda. In the 1896 election, the Democratic candidate William Jennings Bryan gained the support of rural populists, farmers who opposed the gold standard. Bryan represents the first shift in the parties, transforming the Democratic Party into the party of the working class. He wanted inflation to make credit more available to working people. He wanted an income tax so that the rich could be taxed more than the poor (at the time tariffs and excise (sales) taxes provided government revenue, both of which disproportionately hurt the poor). In his famous Cross of Gold speech, Bryan proclaims his vision for the party. In 1896, every corporate executive and their mother donated money to his opponent. Because the Democrats now represented the less prosperous regions of the country (the South, West, and urban immigrant slums) it makes sense that they shifted left on economics. However, Bryan was also very religious; he is famous for fighting against the teaching of evolution in schools. In addition, Democrats remained in total control of the South where they established Jim Crow segregation. Bryan represented a shift in the party on economic issues, but the party remained conservative on social issues.
1900-1940
During the Progressive era 1890s-1920s, all sorts of crazy things happened. There were progressives and anti-progressives in both parties. Teddy Rossevelt was a progressive Republican, Calvin Collidge was not. Woodrow Wilson was a progressive Democrat who created the first income tax and heavily regulated business, but was also a vile racist. This very confusing era ended with FDR and his New Deal in the 1930s. FDR massively increased the size of the government to combat the great depression. Those who supported the New Deal began switching to the Democratic Party. This shift included, for the first time in American history, Northern black voters who were able to overlook the racism of the party because the New Deal provided them with relief during the depression. Because the Republican small government response to the Great Depression was seen as a failure, Democrats became the dominant party during this period. From 1933 until 1994, the Democrats controlled the House of Representatives 58 out of 62 years and the Senate 52 out of 62 years.
1940-1990
However, the massive New Deal Coalition became increasingly strained in the 40s, 50s, and 60s. How could segregationists and African Americans be in the same party? How could rural farmers and urban workers be in the same party? How could anti-Soviet warhawks and socialists be in the same party? The party was divided on multiple occasions between North and South, with Southern Dixiecrats opposing national candidates. In the 1960 election, both candidates, Nixon and Kennedy, were very mealy-mouthed on the issue of Civil Rights, hoping not to offend possible voters on both sides of the issue. However, after Democrats Kennedy and Johsnon decided to support Civil Rights, Republicans employed the so-called Southern Strategy to bring the once Solid Democratic South over to the Republican Party by indirectly appealing to racism. Republicans also appealed to socially conservative Democrats across the country with messages such as law and order. As they brought in these voters, the party itself shifted to the right on social issues. In addition, the Republicans began to appeal to the growing American middle class on issues such as taxes and government spending. As both labor unions and poor white immigrant communities, once bastions of the Democratic party, began to erode in strength and numbers, the children of the members of these Democratic groups became Republicans. Together, this new coalition allowed the Republican party to emerge from its decades-long slump with a new economically and socially conservative vision under Ronald Reagan. On the other hand, Democrats used their new commitment to Civil Rights to appeal to the increasing number of nonwhite voters in the electorate.
I want to emphasize a few things here:

1. The parties were less polarized in the past than they are today. Many Republicans supported Civil Rights such as George Romney (the father of Mitt Romney) and many Democrats opposed Civil Rights (almost every single Democrat in the South).

2. The labels of conservative and liberal as we know them today didn’t exist back then. It made perfect sense at the time to be a racist southern Democrat who believed that the government should do more to help struggling people (southern states were and still are poorer than the rest of the country) but should exclude black people from that help.

3. These shifts happen gradually over time; you can’t point to a single election. Even after the major turning points such as 1896, 1932, and 1964, everyone doesn’t immediately switch parties. For example you still had formerly segregationist southerners in congress as Democrats into the 80s and 90s.